Does the AI Director burn a lot of tokens on OpenAI GPT-4o?

Worried about the bill if the AI is calling OpenAI GPT-4o all day.

Not really — it runs on a schedule (default every 15 minutes during the session), not on every bar, and each review is a compact summary of regime + open positions. On a cheaper tier it's typically pennies a day. Stretch the interval in Settings if you want it even lower.